Geometric Submanifolds of Manifolds

这个项目是围绕理解黎曼流形通过子流形。这个主题和这些方法一直是中心黎曼几何自其概念，并有直接联系的问题和方法在几何群论。关于子流形，PI试图进一步研究由固定黎曼n-流形的本原全测地子流形编码的几何数据。一维全测地子流形最简单的例子就是测地长度谱，它已经引起人们的兴趣超过50年了。此外，测地线长度谱与Laplace-Beltrami算子的特征值谱和流形分析有很强的联系。除了这项研究，PI计划调查哪些流形作为固定黎曼流形的全测地子流形出现。两种情况是非紧型局部对称流形和具有n个标记点的亏格g曲线的模空间。在后一种情况下，最突出的是Teichmuller曲线及其相关的基本群Veech群，它们在几何上，代数几何上和动力学上都很有趣。此外，PI寻求固定流形的子流形的障碍物或特殊性质，不仅是为了理解固定流形，也是为了更好地理解等距浸入其他流形的障碍物。最后，PI计划研究几何计数函数的渐近行为，这些函数编码的几何数据，并与几何之外的主题联系在一起。这些函数计数的全测地流形的固定类型作为一个函数的体积和一个基本的实体，这在最简单的情况下是已知的有联系的几何动力学和应用数论。出现在本建议的对象渗透到数学作为基本的例子绑到基本问题和领域。它们不是数学的产物，而是纯科学和应用科学的产物。此外，随着时间的推移，已被证明不仅在数学中，而且在物理学，化学和计算机科学中非常重要。PI希望进一步促进这些重要的联系，不仅直接与具体成果联系，而且传播PI提案的核心思想。

This project is centers around understanding Riemannian manifolds via submanifolds. This topic and these methods have been central in Riemannian geometry since its conception, and have direct ties to problems and methods in geometric group theory. With regard to submanifolds, the PI seeks to further investigate the geometric data encoded by the primitive totally geodesic submanifolds of a fixed Riemannian n-manifold. The simplest case of 1-dimensional totally geodesic submanifolds is nothing more than the geodesic length spectrum and has garnered interest for more than 50 years. In addition, the geodesic length spectrum has strong connections with the eigenvalue spectrum of the Laplace-Beltrami operator and analysis on manifolds. In addition to this study, the PI plans on investigating which manifolds arise as totally geodesic submanifolds of a fixed Riemannian manifold. Two cases are locally symmetric manifolds of non-compact type and the moduli space of genus g curves with n marked points. In the latter case, most prominent are the Teichmuller curves and their associated fundamental groups Veech groups, which are interesting geometrically, algebro-geometrically, and dynamically. In addition, the PI seeks obstructions or special properties of submanifolds of a fixed manifold, not only in an effort to understand the fixed manifold but also to better understand obstructions to isometrically immersing manifolds into other manifolds. Lastly, the PI plans to investigate the asymptotic behavior of geometric counting functions, what geometric data is encoded by these functions, and ties to this topic outside of geometry. These function counting the totally geodesic manifolds of a fixed type as a function of volume and a basic entities, which in the simplest cases are known to have ties to geometric dynamics and applications to number theory.The objects that arise in the present proposal permeate mathematics as fundamental examples tied to basic problems and areas. They were fathered not by mathematics but from pure and applied science. Moreover, over time, have been proven to be centrally important not only in mathematics but in physics, chemistry, and computer science. The PI hopes to foster further these important connections not only directly with specific results but also in disseminating the core ideaology of the PI's proposal.